Remaining Useful Life Estimation of Lithium-Ion Batteries Based on Small Sample Models

Abstract

Accurate prediction of the Remaining Useful Life (RUL) of lithium-ion batteries is essential for enhancing energy management and extending the lifespan of batteries across various industries. However, the raw capacity data of these batteries is often noisy and exhibits complex nonlinear degradation patterns, especially due to capacity regeneration phenomena during operation, making precise RUL prediction a significant challenge. Although various deep learning-based methods have been proposed, their performance relies heavily on the availability of large datasets, and satisfactory prediction accuracy is often achievable only with extensive training samples. To overcome this limitation, we propose a novel method that integrates sequence decomposition algorithms with an optimized neural network. Specifically, the Complementary Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) algorithm is employed to decompose the raw capacity data, effectively mitigating the noise from capacity regeneration. Subsequently, Particle Swarm Optimization (PSO) is used to fine-tune the hyperparameters of the Bidirectional Gated Recurrent Unit (BiGRU) model. The final BiGRU-based prediction model was extensively tested on eight lithium-ion battery datasets from NASA and CALCE, demonstrating robust generalization capability, even with limited data. The experimental results indicate that the CEEMDAN-PSO-BiGRU model can reliably and accurately predict the RUL and capacity of lithium-ion batteries, providing a promising and reliable method for RUL prediction in practical applications.

1. Introduction

As circumstances like global warming, the release of greenhouse gases, and the consumption of fossil fuels receive escalating attention [1,2], the utilization of electricity and other new energy sources has gradually emerged as a key solution to alleviating environmental pollution. Following several decades of development, new energy vehicles and hybrid aerospace vehicles have gradually come to the forefront of public awareness. In these domains, power batteries [3], as a core component of new energy vehicles, are crucial for ensuring the thermal safety and electrochemical performance necessary for the safe operation of vehicles. Furthermore, research suggests that hybrid technology can not only enhance aircraft performance, but also reduce dependence on fossil fuels and lower aircraft weight [4,5]. With the continuous progress of battery technology, energy density is gradually escalating, making future battery technologies a promising solution to tackle the fossil fuel crisis and environmental problems in the aerospace field.

Batteries, serving as a crucial component for the storage of electrical energy, directly exert an influence on the efficient and safe operation of spacecraft and new energy vehicles. Among the various types of batteries used in these applications, lithium-ion batteries [6] stand out for their superior energy density, extended lifespan, and rapid charging abilities, which greatly improve the stability of spacecraft and new energy vehicles. Nevertheless, with the augmentation of charge–discharge cycles, the battery’s capacity gradually deteriorates as a result of irreversible damage to the internal electrode materials. Once the battery reaches its end-of-life (EOL) capacity threshold [7], replacement becomes requisite. The number of cycles a battery undergoes from its initial state to end-of-life significantly influences its operational longevity. Therefore, investigating the lifespan of batteries is essential for improving their reliability.

Current studies focused on predicting the RUL of lithium-ion batteries primarily employ two approaches: model-based methods and data-driven methods [8]. Model-based approaches focus on analyzing the internal structure of lithium-ion batteries and constructing equivalent models that capture the chemical properties of the materials [9,10]. The main prediction techniques include mechanism-based models, empirical models [11], and equivalent circuit models. For example, Hu et al. [12] proposed a moving average estimation model grounded in electrochemical mechanisms, considering factors such as accuracy, computational intensity, prediction horizon, and fault tolerance. Guha et al. [13] employed a fractional-order equivalent circuit combined with a particle filter to estimate the RUL. This approach enabled real-time analysis of the electrochemical impedance spectrum, and provided accurate predictions of the RUL. However, this approach is computationally intensive. Qiao et al. [14] developed a highly accurate prediction method that combines data-driven techniques with advanced machine learning models for estimating the RUL. Despite its precision, this method is complex to implement, and EMD’s end effects can compromise prediction accuracy. Thus, model-based methods face challenges in providing accurate RUL predictions due to the complex physical and chemical changes occurring within lithium-ion batteries and the significant impact of external environmental factors. This complexity underscores an inherent limitation of model-based approaches [15].

The primary data-driven approach involves utilizing neural networks to learn the nonlinear characteristics of the original capacity data from lithium-ion batteries to achieve accurate predictions. Rouhi et al. [16] proposed a method for predicting the RUL utilizing the gated recurrent unit (GRU) model. While the GRU model performs effectively in predicting the remaining useful life of lithium-ion batteries, its prediction accuracy can be constrained by the need to manually set the model’s hyperparameters. To address this, Sedighimanesh M et al. [17] proposed a composite deep learning model was proposed that integrates CNN, LSTM, and GRU with the PSO algorithm. This approach utilizes PSO to optimize the hyperparameters of the composite model.

The capacity of a lithium-ion battery is a critical indicator of its lifespan. Due to the influence of internal chemical reactions and environmental factors during daily use, lithium-ion batteries may exhibit capacity regeneration phenomena [18]. This results in noisy original capacity data, which significantly impacts the accuracy of RUL predictions. Liu et al. [19] employed the Variational Modal Decomposition (VMD) algorithm to address the effects of capacity regeneration, and used a GRU model for RUL estimation. However, VMD decomposition does not fully eliminate the influence of this noise. Cheng et al. [20] developed a hybrid model that integrates empirical mode decomposition (EMD) with a long short-term memory (LSTM) neural network to estimate the state of health (SOH) and predict the RUL of lithium-ion batteries. While the EMD algorithm can mitigate the effects of capacity regeneration, it is still prone to mode aliasing. To address this issue, Zhang et al. [21] applied the CEEMDAN algorithm to process the original data, overcoming mode aliasing and the difficulty of removing white noise inherent in EMD, thus achieving more accurate RUL predictions. Additionally, Tang et al. [22] proposed a bidirectional GRU model for RUL estimation. This network effectively captures battery capacity information in both forward and backward directions, leading to improved RUL estimation accuracy across various types of batteries.

In summary, the CEEMDAN algorithm effectively decomposes raw capacity data from lithium-ion batteries and removes associated noise. The GRU model is particularly effective at capturing extended dependencies in sequential data. The improved GRU-based network, incorporating bidirectional GRU (BiGRU), enhances the ability to capture long-term dependencies in both forward and backward directions. To improve model accuracy impacted by manually set hyperparameters, the PSO algorithm was utilized for hyperparameter optimization. This study proposes a novel approach for predicting the remaining useful life of lithium-ion batteries, termed CEEMDAN-PSO-BiGRU, which integrates CEEMDAN, PSO, and BiGRU. Experimental results from various test groups in an existing dataset demonstrate that the proposed model exceeds the performance of other models, showcasing excellent feasibility and effectiveness.

2. Related Theories

2.1. Sequence Decomposition

CEEMDAN, short for “Complete Ensemble Empirical Mode Decomposition with Adaptive Noise”, is a time-frequency domain analysis approach based on adaptive signal decomposition. CEEMDAN can decompose the original sequence into several Intrinsic Mode Functions (IMFs) and residual components. Unlike the EEMD [23] algorithm, which adds white noise to the original sequence to induce modal aliasing, CEEMDAN adds IMF components to the raw sequence instead of directly adding white noise. This approach resolves the issue that the white noise in EEMD is difficult to remove after being transferred. Consequently, the CEEMDAN algorithm has superior completeness, adaptability, and decomposition efficiency. The calculation principle is as follows:

1.

Adding Gaussian white noise ${\epsilon }_0{\omega }^n\left(t\right)$ to the original signal X(t), $X^n\left(t\right)$ is obtained, where ${\epsilon }_i(i=0,1,2,\cdots ,I)$ denotes the standard deviation of Gaussian white noise, specifically

$$X^n\left(t\right)=X\left(t\right)+{\epsilon }_0{\omega }^n\left(t\right)$$

(1)

2.

Perform EMD decomposition of $x^n\left(t\right)$ and calculate the mean value of the first intrinsic mode function (IMF) to obtain the first mode ${\tilde{m}}_1\left(t\right)$,

$${\tilde{m}}_1\left(\begin{array}{c}t\end{array}\right)={\displaystyle \frac{1}{N}}\sum _{n=1}^N{E}_1{x}^n\left(\begin{array}{c}t\end{array}\right)$$

(2)

3.

Calculate the residual generated from the first decomposition $r_1\left(t\right)$,

$$r_1\left(\begin{array}{c}t\end{array}\right)=x\left(\begin{array}{c}t\end{array}\right)-{\tilde{m}}_1\left(\begin{array}{c}t\end{array}\right)$$

(3)

4.

Add noise ${\epsilon }_1{E}_1{\omega }^n\left(t\right)$ to the residual $r_1\left(t\right)$ to create a new signal. Perform EMD decomposition on this new signal to obtain the second IMF, and then calculate its average to derive the second mode ${\tilde{m}}_2\left(t\right)$,

$${\tilde{m}}_2\left(t\right)={\displaystyle \frac{1}{N}}\sum _{n=1}^N{E}_1\left(r_1\left(t\right)+{\epsilon }_1{E}_1{\omega }^n\left(t\right)\right)$$

(4)

5.

Calculate the value of the residual generated from the second decomposition,

$$r_2\left(\begin{array}{c}t\end{array}\right)=r_1\left(\begin{array}{c}t\end{array}\right)-{\tilde{m}}_2\left(\begin{array}{c}t\end{array}\right)$$

(5)

6.

Repeat steps 4 and 5 to obtain the kth mode ${\tilde{m}}_{\widehat{\kappa }}\left(t\right)$ and the kth residual $r_k\left(\begin{array}{c}t\end{array}\right)$. If the termination condition is met at this point, then

$$\begin{array}{cc}& {\tilde{m}}_{\dot{k}}={\displaystyle \frac{1}{N}}\sum _{n=1}^k{E}_1\left(r_{\dot{k}-1}\left(t\right)+{\epsilon }_{\dot{k}-1}{E}_{\dot{k}-1}{\omega }^n\left(t\right)\right)\hfill \\ & r_k\left(t\right)=r_{k-1}\left(t\right)-{\tilde{m}}_{\dot{k}}\left(t\right)\hfill \end{array}$$

(6)

The original signal can finally be expressed as

$$x\left(\begin{array}{c}t\end{array}\right)=\sum _{k=1}^K{\tilde{m}}_k\left(\begin{array}{c}t\end{array}\right)+r_k\left(\begin{array}{c}t\end{array}\right)$$

(7)

2.2. Particle Swarm Optimization

The Particle Swarm Optimization (PSO) algorithm [24,25], introduced by Kennedy and Eberhart in 1995, is inspired by the flocking behavior of birds, fish, and other animals searching for food in groups. The PSO algorithm offers several advantages. Firstly, unlike gradient-based algorithms that heavily depend on the initial positions of particles in the solution space, PSO’s performance is less sensitive to the initial positions. Secondly, PSO does not require the gradient of the objective function to be defined across the entire solution space. Lastly, previous studies [26,27] have shown that PSO has relatively simple code, and is efficient in finding suboptimal or global optimal solutions. Therefore, this study uses the PSO to optimize the hyperparameters of the BiGRU model for predicting the RUL. The calculation steps are as follows:

1.

Initialization:

• Generation of the particle swarm: randomly initialize a set of particles within the defined search space, with each particle representing a possible solution.
• Initialization of velocity: assign a random initial velocity to each particle.
• Initial position: ascertain the initial position of each particle.
• Initialization of individual best solution: the initial position of each particle is designated as its current individual best solution.
• Initialization of global best solution: choose the current best solution among all individual best solutions as the global best solution.

2.

Iterative update: Perform the following operations for each particle, repeating the iterations multiple times until the termination condition is met:

• Velocity update.

Update the velocity of the particle, which determines the direction and step size of the particle’s movement in the next iteration. The velocity update formula is:

$${\nu }_i\left(\begin{array}{c}t+1\end{array}\right)=cos·{\nu }_i\left(\begin{array}{c}t\end{array}\right)+c_1·r_1·\left(\begin{array}{c}pBes{t}_i-x_i\left(\begin{array}{c}t\end{array}\right)\end{array}\right)+c_2·r_2·\left(\begin{array}{c}gBest-x_i\left(\begin{array}{c}t\end{array}\right)\end{array}\right)$$

(8)

where $v_i\left(t+1\right)$ represents the velocity of particle i at iteration $t+1$; $v_i\left(t\right)$ represents the velocity of particle i at iteration i; $\omega$ represents the inertia weight; $c_1,c_2$ represent the acceleration factors; $r_1,r_2$ are random numbers within the range $\left[\begin{array}{c}0,1\end{array}\right]$; pBest represents the historical best position of particle i; gBest represents the global historical best position; and $x_i\left(t\right)$ represents the position of particle i at iteration i.

• Position update.

Calculate the new position of the particle based on the updated velocity:

$$x_i\left(t+1\right)=x_i\left(t\right)+v_i\left(t+1\right)$$

(9)

where $x_i\left(t+1\right)$ represents the position of particle i at iteration $t+1$; $x_i\left(t\right)$ represents the position of particle i at iteration i; and $v_i\left(t+1\right)$ represents the velocity of particle i at iteration $t+1$. An update schematic is shown in Figure 1.

• Evaluate fitness.

Assess the fitness of each particle at its new position. The fitness function typically represents the goal of the optimization problem, indicating the quality of the potential solution.

• Update pBest and gBest.

  1.

  Update pBest: if the current fitness value of a particle surpasses its previous best, adjust the particle’s pBest accordingly;

  2.

  Update gBest: if a particle’s current fitness value exceeds the global best, adjust the gBest accordingly;

  3.

  Termination condition: The algorithm concludes either when the maximum number of iterations is reached or when the gBest meets the specified accuracy requirements. The gBest is then considered the optimal solution to the problem.

3.

Output results.

The final output is the gBest along with the corresponding optimal fitness value.

2.3. Bidirectional Gated Recurrent Unit

The Gated Recurrent Unit (GRU) [28] is a variant of the Long Short-Term Memory (LSTM) network [29]. Given the strong temporal correlations in lithium-ion battery performance, selecting an appropriate time series prediction model is essential. While Recurrent Neural Networks (RNNs) can capture nonlinear relationships in time series data, they face issues such as vanishing or exploding gradients due to their time dependency. LSTM networks address these issues with their specialized architecture, which includes an input gate, a forget gate, an output gate, and a cell state, allowing for effective long-term memory retention. However, this complexity results in lengthy training times. In contrast, the GRU simplifies the LSTM structure by primarily using a reset gate and an update gate. This simplification improves training efficiency while preserving prediction accuracy. A schematic diagram of a GRU unit is illustrated in Figure 2. The mathematical formulas for GRU are as follows:

$$r_t=\sigma \left(W_r·\left[h_{t-1},x_t\right]\right)$$

(10)

$${\tilde{h}}_t=tanh\left(W·\left[r_t\odot h_{t-1},x_t\right]\right)$$

(11)

$$z_t=\sigma \left(W_z·\left[h_{t-1},x_t\right]\right)$$

(12)

$$h_t=\left(\begin{array}{c}1-z_t\end{array}\right)\odot {\tilde{h}}_t+z_t\odot h_{t-1}$$

(13)

where $r_t$ and $z_t$ represent reset gate and update gate, respectively. ${\tilde{h}}_t$ and $h_t$ represent the candidate hidden state and the update hidden state, respectively. $W_r$, $W_z$ and W, respectively, represent the weight parameters of the GRU model. $\sigma$ is the sigmoid activation function. ⊙ represents the dot product operation.

As shown in Figure 2, in the traditional GRU model, information state propagation occurs only from the past to the future. Given that predicting the RUL of lithium-ion batteries relies heavily on time series information, this paper employs two GRU models with opposite propagation directions to form a Bidirectional GRU (BiGRU) model [30]. This approach facilitates the extraction of temporal information from both forward and backward directions. The structure of the BiGRU model is shown in Figure 3.

2.4. The Proposed Model

Building on the strengths of the aforementioned components, we propose an integrated RUL prediction model, named CEEMDAN-PSO-BiGRU. The process involves the following steps:

1.

Feature extraction: the CEEMDAN algorithm is employed to extract feature components from the original capacity sequence;

2.

Hyperparameter optimization: the PSO algorithm is used to perform hyperparameter tuning for the BiGRU model;

3.

Model training: the optimized BiGRU model is then trained using the extracted components;

4.

RUL estimation: the trained BiGRU model is then utilized to precisely estimate the RUL of lithium-ion batteries.

A schematic diagram of the prediction process is show in Figure 4. The steps involved are as follows.

First, preprocess the original data by checking for missing values. If any are found, fill them with the average of the preceding and following values to ensure the dataset is complete.

Next, the CEEMDAN algorithm is applied to decompose the processed capacity data into IMFs of various frequencies.

Then, standardize the obtained components by normalizing the data to a range of (0, 1) to facilitate neural network processing.

Following this, use the PSO algorithm to identify the optimal hyperparameters for the model. Train the model for each component separately using the optimal number of GRU units and learning rate parameters determined by the PSO optimization. An early stopping strategy is employed to prevent overfitting.

Finally, develop a comprehensive RUL prediction model that integrates the predictions from each component and the residual component to produce the final RUL estimate.

3. Experimental Setup

3.1. Experimental Dataset

This study utilizes eight publicly available datasets to evaluate the effectiveness of the proposed model. The datasets include B0005, B0006, B0007, and B0018 from NASA, as well as $CS{2}_{-}35,CS{2}_{-}36,CS{2}_{-}37$, and $CS{2}_{-}38$ from CALCE. Both NASA and CALCE employ 18650 lithium-ion battery models. The anode is primarily composed of graphite, while the cathode typically consists of lithium cobalt oxide (LiCoO₂) lithium iron phosphate (LiFePO₄), or nickel-cobalt-manganese oxide (NCM). Notably, lithium iron phosphate is recognized for its excellent thermal stability and safety, making it particularly suitable for electric vehicles and energy storage applications. The electrolyte in 18650 batteries generally consists of lithium salts dissolved in organic solvents to ensure optimal ionic conductivity.

The capacity degradation data of lithium-ion batteries exhibit pronounced nonlinearity and non-stationarity.

Table 1. Battery information for NASA and CALCE datasets.

Dataset Name	Capacity (Ah)	Threshold (Ah)	Temperature (°C)	Discharge Current (A)	Cutoff Voltage (V)
NASA	2	1.4	24	2	2.7, 2.5, 2.2, 2.5
CALCE	1.1	0.77	1	1.1	2.7, 2.5, 2.2, 2.5

summarizes the key parameters of the two battery datasets, while Figure 5a and b display the capacity degradation curves for NASA and CALCE batteries, respectively. The red threshold line indicates the point at which the battery should be retired, specifically when its capacity falls to 70% of the original value. At this stage, the battery’s energy storage capability is significantly diminished, resulting in a substantial reduction in device runtime and negatively impacting user experience. Furthermore, the battery may pose safety risks, such as overheating, swelling, or short circuits. As the degradation progresses, the internal structure of the battery may become unstable, increasing the risk of failure.

3.2. CEEMDAN Decomposition Results

The overall capacity of lithium-ion batteries typically exhibits a decreasing trend throughout their usage cycle. However, factors such as capacity regeneration during charging and environmental influences introduce significant noise into the original capacity data. To resolve this problem, this study employs the CEEMDAN algorithm to analyze the lithium-ion capacity data, extracting multiple IMFs and one residual component to mitigate the impact of noise. IMF1 and IMF2 represent short-term fluctuations caused by capacity regeneration during the degradation process, which are classified as noise. Removing these components can enhance the smoothness of the original data and reduce interference. Conversely, IMF3 corresponds to periodic fluctuations, revealing the performance changes of the battery under specific operating conditions. For instance, the decomposition results for NASA’s B0005 battery are illustrated in Figure 5d.

Figure 5c shows that the denoised data closely matches the actual data, indicating a consistent trend in capacity decline. Thus, the capacity data obtained after CEEMDAN decomposition can be considered effective for forecasting the RUL. To further verify the reliability of the data, this paper will use the Pearson correlation coefficient to assess the correlation between the original capacity data and the denoised data. The formula for calculating the Pearson correlation coefficient is as follows:

$$r={\displaystyle \frac{{\sum }_{i=1}^n\left(X_i-\overline{X}\right)\left(Y_i-\overline{Y}\right)}{\sqrt{{\sum }_{i=1}^n{\left(X_i-\overline{X}\right)}^2}\sqrt{{\sum }_{i=1}^n{\left(Y_i-\overline{Y}\right)}^2}}}$$

(14)

where $X_i$ represents the original capacity data, ${\overline{X}}_i$ represents the mean value of the original capacity data, $Y_i$ represents the denoised capacity data, ${\overline{Y}}_i$ represents the average value of the denoised capacity data, and r represents the Pearson correlation coefficient.

The Pearson’s r ranges from −1 to 1. An absolute value of r greater than 0.7 signifies a strong linear relationship. When r falls between 0.3 and 0.7, the linear relationship is considered moderate. When r is below 0.3, the linear relationship is weak. A value of r closer to 0 reflects a weaker linear association between the two variables.

The Pearson’s r between the original capacity data from the public lithium-ion battery datasets used in this paper and the capacity data processed by CEEMDAN is calculated through experiments, as shown in

Table 2. The Pearson correlation coefficient between the denoised values and the original values.

Battery Model	Pearson Correlation Coefficient
B0005	99.7%
B0006	97.3%
B0007	99.7%
B0018	97.7%
$CS{2}_{-}35$	97.3%
$CS{2}_{-}36$	99.7%
$CS{2}_{-}37$	98.7%
$CS{2}_{-}38$	97.7%

.

Table 2 illustrates that the Pearson’s r between the original capacity data from the public lithium-ion battery datasets used in this paper and the capacity data processed by CEEMDAN is calculated through experiments, as shown in Table 2. Between the denoised capacity data and the original capacity data from both the NASA and CALCE datasets, it exceed 97%. This indicates that the data following decomposition effectively reflects the trend of capacity attenuation in lithium-ion batteries while mitigating the impact of capacity regeneration. Figure 5d reveals that, although the magnitude of the Intrinsic Mode Functions (IMFs) is very small, the capacity regeneration phenomenon is limited. However, ignoring this phenomenon could affect the accuracy of the RUL prediction model. Therefore, the introduction of the CEEMDAN algorithm in this paper is crucial for addressing capacity regeneration, ensuring that the BiGRU model can accurately predict the RUL of lithium-ion batteries.

3.3. Experimental Results Evaluation Indices

To evaluate the constructed RUL prediction model, the mean absolute error (MAE) and root mean square error (RMSE) are employed as the model evaluation indicators proposed herein. The calculation formulas are as follows:

$$MAE={\displaystyle \frac{1}{n}}\sum _{t=1}^n∥x_t-{\widehat{x}}_t∥$$

(15)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{t=1}^n{\left(x_t-{\widehat{x}}_t\right)}^2}$$

(16)

where n represents the number of capacity data of the experimental lithium-ion battery, $x_t$ represents the real value of the capacity of the experimental lithium-ion battery, and ${\widehat{x}}_t$ represents the predicted value of the lithium-ion battery.

4. Experimental Results and Analysis

4.1. Methods

To assess the effectiveness of the proposed model, three models—GRU, BiGRU, and CEEMDAN-GRU—were introduced for comparative experiments using capacity datasets from four battery types (NASA’s B0005, B0006, B0007, and B0018). Furthermore, to test the model’s generalization ability when the dataset has a small number of samples, different proportions of the training set were used while keeping the total dataset the same. Specifically, the training sets were divided into 50% and 60%, where the 85th and 101st cycles were the first predicted values, and for B0018, the 67th and 80th cycles were the first predicted values. The MAE and RMSE values for each method were calculated and compared. To improve the model’s learning capability on the data, a sliding window was introduced to divide the dataset. Each time, 30% of the training data were used as the training set, and with each slide of the window, 10% of the window size was shifted. Additionally, to further substantiate the generalization capability of the proposed model, the identical settings were implemented on the CALCE $CS{2}_{-}35, CS{2}_{-}36, CS{2}_{-}37, CS{2}_{-}38$ datasets to further validate the model’s feasibility and superiority.

4.2. NASA Battery Dataset Experiments

4.2.1. 60% Data as Training Set

In this experiment, 60% of the data were used for training, while the remaining 40% was allocated for testing. For the B0018 dataset, predictions began at the 79th cycle, whereas for the other three datasets, predictions started from the 100th cycle. Figure 6 presents the prediction results for the four battery groups. To offer a more intuitive assessment of the reliability of the experimental results, the evaluation metrics described in Section 3.3 were also used. The MAE and RMSE values for the four lithium-ion battery groups are presented in

Table 3. Prediction index results of NASA 60% dataset as prediction training set.

Battery Model	Training Set Ratio	Evaluation Index	GRU	BiGRU	CEEMDAN-GRU	CEEMDAN-BiGRU	CEEMDAN-PSO-BiGRU
B0005	60%	MAE	0.01121	0.01188	0.00847	0.00632	0.00632
		RMSE	0.01804	0.01696	0.01226	0.00110	0.00901
B0006	60%	MAE	0.02024	0.02061	0.01568	0.01413	0.01098
		RMSE	0.03168	0.03030	0.02046	0.01856	0.01568
B0007	60%	MAE	0.00940	0.00994	0.00790	0.00773	0.00613
		RMSE	0.01576	0.01484	0.01090	0.01077	0.00826
B0018	60%	MAE	0.01544	0.01806	0.01380	0.01306	0.01260
		RMSE	0.02669	0.02708	0.01975	0.02057	0.01958

.

As shown in Figure 6 and Table 3, the proposed model demonstrates lower MAE and RMSE values compared to other methods, highlighting the effectiveness and accuracy of the CEEMDAN-PSO-BiGRU model. For example, in Figure 6, when 60% of the data are used for training, the larger dataset allows the model to better capture the characteristics of capacity degradation. Consequently, the predictions closely match the actual results, and the high level of curve fitting reflects the model’s accuracy in forecasting the RUL.

4.2.2. 50% Data as Training Set

For the original capacity data of lithium-ion batteries, the experiment uses half of the data for training and the other half for testing. For the B0018 group, predictions start from the 66th cycle, while for the other three groups, predictions begin from the 84th cycle. Figure 7 presents the prediction results for the four battery groups.

To assess the reliability of the prediction results, the evaluation indicators in Section 4.3 are employed as the judgment criteria for the experimental outcomes. The MAE and RMSE of the four groups of batteries are presented in Table 3.

As can be perceived from Figure 7 and

Table 4. Prediction index results of NASA 50% dataset as prediction training set.

Battery Model	Training Set Ratio	Evaluation Index	GRU	BiGRU	CEEMDAN-GRU	CEEMDAN-BiGRU	CEEMDAN-PSO-BiGRU
B0005	50%	MAE	0.01924	0.02177	0.01721	0.00692	0.00692
		RMSE	0.02521	0.02808	0.01409	0.01968	0.01025
B0006	50%	MAE	0.03603	0.04021	0.01808	0.01542	0.01173
		RMSE	0.04359	0.04791	0.02324	0.02131	0.01946
B0007	50%	MAE	0.01686	0.01903	0.00997	0.01145	0.00638
		RMSE	0.02194	0.02437	0.01279	0.01401	0.00923
B0018	50%	MAE	0.02419	0.02685	0.01431	0.01969	0.01235
		RMSE	0.03287	0.03549	0.02121	0.02433	0.01977

, in the presence of the capacity regeneration phenomenon, the overall prediction trends of several models are essentially the same, yet the deviations between the predicted values of the other three methods and the real values are relatively substantial, among which GRU and BiGRU perform the most poorly. Taking the B0007 battery as an example, it is evident from the figure that the predictions made by the proposed model closely align with the actual values. Additionally, the prediction metrics are lower compared to those of the other three methods, demonstrating the model’s robustness and generalization ability even with limited data.

4.3. Experiments on CALCE Battery Dataset

To further validate the generalization capabilities of different models and the prediction improvements across various data scales, four sets of experimental data—$CS{2}_{-}35$, $CS{2}_{-}36$,$CS{2}_{-}37$, and $CS{2}_{-}38$—are selected. Unlike NASA’s dataset, CALCE’s dataset contains a significantly higher number of battery data cycles. Consequently, the data exhibits increased noise from the capacity regeneration phenomenon, which more effectively demonstrates the generalization ability of the proposed model. The number of cycles for the four lithium-ion battery groups from CALCE and their corresponding prediction starting points are summarized in

Table 5. Cycle numbers and prediction starting points of CLACE batteries.

Battery Model	Cycles	Training Set Ratio	Prediction Start
$CS{2}_{-}35$	882	60%	530
		50%	442
$CS{2}_{-}36$	936	60%	562
		50%	469
$CS{2}_{-}37$	972	60%	584
		50%	487
$CS{2}_{-}38$	996	60%	598
		50%	499

.

4.3.1. 60% Data as Training Set

Select 60% of the data samples to constitute the training set, and the remaining 40% to form the test set. The prediction outcomes of different models are presented in Figure 8. In accordance with the NASA experiment operation, the evaluation indicators in Section 4.3 are introduced as the judgment criteria for the experimental results. The MAE and RMSE of the four groups of batteries are presented in

Table 6. Prediction index results of CALCE 60% dataset as prediction training set.

Battery Model	Training Set Ratio	Evaluation Index	GRU	BiGRU	CEEMDAN-GRU	CEEMDAN-BiGRU	CEEMDAN-PSO-BiGRU
$CS{2}_{-}35$	60%	MAE	0.01104	0.00865	0.00863	0.01249	0.00432
		RMSE	0.01775	0.01468	0.01151	0.01417	0.00762
$CS{2}_{-}36$	60%	MAE	0.01881	0.00969	0.00884	0.01109	0.00512
		RMSE	0.01294	0.01511	0.01245	0.01428	0.00826
$CS{2}_{-}37$	60%	MAE	0.01099	0.00801	0.00713	0.01245	0.00484
		RMSE	0.01670	0.01354	0.01062	0.01496	0.00910
$CS{2}_{-}38$	60%	MAE	0.01217	0.00839	0.00919	0.01475	0.00760
		RMSE	0.01683	0.01389	0.01291	0.01708	0.01097

.

As can be observed from Figure 8 and Table 6, when 60% of the data are employed as the training set, the MAE and RMSE of CALCE batteries are significantly smaller than those of NASA batteries. This is because the number of dataset samples of CALCE is much larger than that of NASA datasets. When the dataset samples of NASA are insufficient, the characteristics of capacity degradation are not well captured, resulting in the model prediction accuracy being inferior to that of CALCE datasets. Taking $CS{2}_{-}35$ in Figure 8 as an example, it can be perceived that the GRU and BiGRU models have the poorest prediction effect and cannot handle the capacity regeneration phenomenon effectively. Since the CEEMDAN-GRU introduces the CEEMDAN algorithm, it can alleviate the capacity regeneration phenomenon. However, since the capacity degradation data has a strong time correlation, the GRU unit cannot completely capture these time series information, resulting in a prediction accuracy inferior to the model proposed in this paper.

4.3.2. 50% Data as Training Set

Select 50% of the data samples as the training set, and the remaining 50% as the test set. The results of different models are show in Figure 9. At the same time, it is consistent with the experimental operation of using 60% as the training set. The evaluation indicators in Section 4.3 are introduced as the judgment criteria for the experimental results. The MAE and RMSE of the four groups of batteries are shown in

Table 7. Prediction index results of CALCE 50% dataset as prediction training set.

Battery Model	Training Set Ratio	Evaluation Index	GRU	BiGRU	CEEMDAN-GRU	CEEMDAN-BiGRU	CEEMDAN-PSO-BiGRU
$CS{2}_{-}35$	50%	MAE	0.02150	0.01955	0.00998	0.00659	0.00621
		RMSE	0.03004	0.02837	0.01237	0.01147	0.01029
$CS{2}_{-}36$	50%	MAE	0.02431	0.02249	0.00874	0.00764	0.00579
		RMSE	0.03447	0.03252	0.01246	0.01066	0.00994
$CS{2}_{-}37$	50%	MAE	0.02149	0.01956	0.01235	0.00735	0.00601
		RMSE	0.03018	0.02840	0.01484	0.01272	0.01068
$CS{2}_{-}38$	50%	MAE	0.02111	0.01922	0.01709	0.01253	0.00369
		RMSE	0.02899	0.02729	0.01884	0.01578	0.00636

.

As can be discerned from the figure and the table, in contrast to the 50% data experiment of CALCE, the prediction accuracy errors of the other three methods increase. This is because the data capacity sample of CALCE becomes smaller, and the GRU and BiGRU models have difficulty in capturing more time-series features. In addition, the phenomenon of capacity regeneration begins to occur in large quantities at the prediction starting point, resulting in poorer prediction accuracy. However, the data processed by the CEEMDAN algorithm largely alleviates the capacity regeneration phenomenon, enabling the PSO-BiGRU model to handle the characteristics of time series better. Thus, when the data sample is reduced, the accuracy of time series prediction improves, further demonstrating the advantages of the model presented in this study.

4.4. Comparison with Other Methods

To better demonstrate the superiority of the method proposed in this study for predicting the RUL of lithium batteries, we selected and compared the prediction performance of other methods under the same battery dataset and similar starting points. For details, see

Table 8. Comparison with other methods.

Battery Model	Method	Initial Prediction Cycle Point	RMSE	MAE
$CS{2}_{-}35$	CEEMDAN-PSO-BiGRU	442	0.01029	0.00621
		530	0.00762	0.00432
	CNN-DBLSTM	441	0.02861	0.02089
		529	0.02352	0.01703
	Hybrid Model	442	0.02580	0.01350
		-	-	-
$CS{2}_{-}36$	CEEMDAN-PSO-BiGRU	442	0.00994	0.00579
		530	0.00826	0.00512
	CNN-DBLSTM	-	-	-
		-	-	-
	Hybrid Model	442	0.02030	0.01570
		-	-	-
$CS{2}_{-}37$	CEEMDAN-PSO-BiGRU	442	0.01068	0.00601
		530	0.00910	0.00484
	CNN-DBLSTM	-	-	-
		-	-	-
	Hybrid Model	442	0.01920	0.01360
		-	-	-
$CS{2}_{-}38$	CEEMDAN-PSO-BiGRU	442	0.00636	0.00369
		530	0.01097	0.00760
	CNN-DBLSTM	441	0.01726	0.01292
		529	0.03323	0.02736
	Hybrid Model	442	0.01910	0.01360
		-	-	-

.

It can be derived from Table 8 that when the prediction starting points are similar, the method presented in this paper has higher prediction accuracy than the Hybrid Model [31] and CNN-DBLSTM [32]. Among them, when the prediction starting point is smaller, the approach proposed in this study demonstrates higher accuracy than CNN-DBLSTM. This indicates that CEEMDAN-PSO-BiGRU has a notable advantage in scenarios with small sample sizes.

5. Conclusions

This paper introduces a lithium-ion battery RUL prediction approach that integrates CEEMDAN decomposition, the PSO optimization algorithm, and the BiGRU model. CEEMDAN is utilized to decompose the original capacity data, effectively mitigating the capacity regeneration phenomenon. The critical hyperparameters of the BiGRU model are then optimized using the PSO algorithm, which significantly improves the model’s prediction accuracy and generalization capability. The main conclusions of this paper can be summarized as follows:

1.

This approach takes the capacitance regeneration phenomenon into account and utilizes the CEEMDAN algorithm to decompose the processed capacity data into IMFs of various frequencies. In comparison with other methods, the prediction results of the combined model exhibit lower RMSE and error. Consequently, the application of decomposition algorithms is conducive to enhancing the prediction performance.

2.

The PSO algorithm optimizes the key hyperparameters of the BiGRU. The experimental results show that using the PSO algorithm can reduce MAE and RMSE, thereby improving the prediction accuracy of the model.

3.

BiGRU can capture the complex long-term and short-term dependencies in time series, enhance the model’s sensitivity to changes in battery life trends, can deal with the nonlinear and non-stationary problems in lithium battery RUL prediction more effectively, and has high application value and practical feasibility. The method proposed in this paper offers new ideas and technical routes for lithium battery RUL prediction and has broad promotion prospects in practical applications such as battery management systems.

In summary, this study investigated the application of the CEEMDAN-PSO-BiGRU algorithm for predicting the RUL of batteries, and found that this approach significantly enhances prediction accuracy. To further strengthen the integrity of this research, future studies should explore specific real-world applications of this predictive technology. For instance, in energy storage systems, this technology can optimize energy management and improve system efficiency; in the electric vehicle sector, precise battery state predictions can extend driving range and enhance user experience; and in battery management systems, this approach can facilitate more effective charge and discharge strategies, ensuring safety and prolonging battery life. These applications not only underscore the practical relevance of the predictive technology, but also offer a promising outlook for its widespread adoption across various industries.